Determination of the third- and fifth-order optical nonlinearities:
the general case
V. Besse•G. Boudebs• H. Leblond
Received: 3 October 2013 / Accepted: 22 January 2014 ÓSpringer-Verlag Berlin Heidelberg 2014
Abstract We compute the evolution of the intensity (I) and the phase (u) of a beam propagating in a nonlinear (NL) isotropic medium exhibiting third- and fifth-order NL optical characteristics. All formulas are analytic, but the general case requires a numerical inversion by means of Newton’s method. The solutions may differ if some coef- ficients vanish, so they are given in all cases up to the fifth- order nonlinearities. The analytical relations allow us to fit the experimental data using the recently introduced D4r- Z-scan method. Carbon disulfide is tested at 532 and 1,064 nm in the picosecond regime deducing NL coeffi- cients related to third- and fifth-order optical susceptibilities.
1 Introduction
Large nonlinearities are essential for many applications in optics. For example, highly nonlinear (NL) absorbing materials can find applications in the domain of optical limiting [1], and large NL refractive index is used for ultrafast all-optical switching [2–4]. Another application is the generation and the propagation of stable two- or three- dimensional dissipative solitons [5], theoretically predicted and experimentally achieved [6] with specific third-order and fifth-order optical nonlinearities. For that, it is neces- sary to accurately measure the NL coefficients especially
for different experimental parameters. Many methods allow to measure third-order NL refraction and two-photon absorption (2PA) [7,8]. In Ref. [8], we have shown using D4rmethod that the numerical calculations allow to obtain simple relations that can be used for the measurements simplifying the procedure especially for NL absorbing material. We will show hereafter that using this method, one can determine high-order susceptibilities in all cases.
Previously, we have solved analytically, in Refs. [9] and [10] with adequate approximation, the differential equation that governs the evolution of the intensity in the presence of n2, the nonlinear index coefficient and both 2PA and three photon one (3PA). Using NL fitting combined to Runge–Kutta numerical integration method, we have numerically solved the beam propagation equation.
Another approach was used in [11] in the case of third- (n2) and fifth-order (n4) NL refractive indices with 2PA only.
Here, we consider the general case with materials exhibiting all the NL coefficients up to the fifth-order including 2PA, 3PA,n2andn4.
The paper is organized as follows. In Sect.2, we present the analytical solution of the differential equations that govern the evolution of the intensity and the NL phase shift. We consider each possible case depending on the possible vanishing of one or several of the coefficients.
Newton’s method is used in order to numerically inverse the analytical expression which describes the evolution of the intensity during the propagation. With this result, we can obtain the analytical expression of the NL phase shift related to the third- (n2) and fifth-order (n4) NL refractive indices. In Sect.3, we validate our solution by determining the NL absorption coefficient and refractive index of a well-known reference NL liquid: carbon disulfide (CS2), in the picosecond regime at 1,064 and 532 nm using the D4r- Z-scan method.
V. Besse (&)G. BoudebsH. Leblond LUNAM Universite´, Universite´ d’Angers, LPhiA, Laboratoire de Photoniques d’Angers, EA 4464, 49045 Angers Cedex 01, France
e-mail: [email protected];
[email protected] DOI 10.1007/s00340-014-5777-2
2 Theory
We consider a beam propagating along the z-axis in a sample exhibiting linear absorption [coefficient a (m-1)], 2PA [coefficient b (m/W)], 3PA [coefficient c (m3/W2)], Kerr effect [third-order NL refractive indexn2(m2/W)] and fifth-order NL refractive index [n4 (m4/W2)]. Under the slowly varying envelope and thin sample approximations, the optical intensityI(GW/cm2) and the phaseusatisfy the equations
dI
dz¼ aIbI2cI3; ð1Þ
du
dz ¼k n 2Iþn4I2
; ð2Þ
wherek=2p/kis the modulus of the wave vector andkis the wavelength. We consider here the general case where each coefficient is nonzero. The differential Eq. (1) governs the evolution of the intensity as a function of the propagation distance z in the medium. The sample is located between the planez=0 and the planez=L. The boundary conditions are I(z=0) =I0 andI zð ¼LÞ ¼IL. It is possible to exactly solve this equation by separating the variables. After partial fraction decomposition,
dz¼ 1
cXþXIþ 1
cXðXXþÞðIXÞ
1
cXþðXXþÞðIXþÞ
dI;
ð3Þ
where X¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b24ac p b
.
ð Þ2c (assuming that b2[4ac), integration of Eq. (3) fromz=0 toLleads to
z¼ ln IIL
0
cXþX
þ
ln IILX
0X
cXðXXþÞ
ln IILXþ
0Xþ
cXþðXXþÞ 2
4
3 5: ð4Þ Expression (4) can be inverted numerically using Newton’s method. Let us recall that Newton’s method is iterative and based on the approximation of the function z(I) using a first-order Taylor expansion as
z Ið Þ ¼z IL;0
þz0 IL;0
IL;1IL;0
; ð5Þ
where the prime holds for the derivative with respect to I. Solving the equationz Ið Þ ¼Lwithz(I) given by Eq. (5) yields the recurrence relation IL;nþ1 ¼IL;n
zðIL;nÞ L
=z0ðIL;nÞwhich is used in the iterative process.
Newton’s method is easy to implement but requires an adequate choice of the starting values, which must be close enough to the final solution in order to converge. It can be applied provided that the derivative can be computed, which is easy sincez(I) has the explicit expression (4) and does neither vanish nor go to infinity.z0ðIÞ ¼0 corresponds
to an infinite absorption, which is unphysical. In contrast, z0ðIÞ ! 1 corresponds to the non-absorbing case, which may occur for the self-induced transparency but taking into account our experimental parameters and results, this situ- ation did not happened into CS2. In order to avoid unphysical solutions, it is important to properly choose the initial valueIL,0as close as possible to the exact roots. For smallL, Eq. (1) can be solved approximately as
IL;0¼I0aI0þbI02þcI03
L: ð6Þ
We use this approximate value to initiate Newton’s method.
Then, the NL phase shift Du is obtained by solving Eq. (2) and using Eq. (1),
du
dI ¼k nð 2Iþn4I2Þ
aIþbI2þcI3 : ð7Þ
After decomposing its right-hand side member in partial fractions, Eq. (7) is integrated from I0toILto yield
Du¼ k
cðXþXÞ ðn2þn4XþÞln ILXþ I0Xþ
þðn2þn4XÞln ILX
I0X
:
ð8Þ
Here, Du¼uLu0 is the NL phase shift due to the propagation through the NL medium. We replaced IL by the valueIL,nthat we found previously. By fitting Z-scan or D4rtraces, we should be able to determine experimentally the value of each coefficient:a,b,c[with Eq. (4)],n2and n4[with Eq. (8)].
Note that Eqs. (4) and (8) are valid only ifa,bandcare nonzero. If this condition is not satisfied, direct solution of Eqs. (1) and (2) is required. The various cases and corre- sponding solutions are reported in Tables1 and 2 (see Appendix).
In next section, we describe our experimental setup and results with a cell filled with CS2. We demonstrate the reliability of the method by fitting the experimental results and determining the corresponding coefficients.
3 Experimental results
The experimental data were obtained using D4r method inside a Z-scan 4f-system (see Fig.1). The sample is set on a translation stage moving around the focus position (Z=0). All the process is controlled by a computer allowing, for each step of the motor, to open a shutter (not shown in the figure) and to acquire an image of the output pulse via a CCD camera. The second arm is used to monitor the energy fluctuation of the incident laser beam.
The 4fimaging system in the linear regime can be merely described using a simple model based on Fourier optics
(see Ref. [12]). The object used to characterize the NL medium is placed before the lens L1, where any beam profiles can be used. We considered a circular object (top hat beam) with an optimized radius in order to obtain a Rayleigh range larger than the sample thickness and smaller than the motor range. The self-diffracted spectrum on the induced instantaneous phase shift gives rise to changes in the transmitted intensity then into the mea- surement of the beam waist in the image plane while moving the sample around the focal region. Note that the lens L2contributes to produce the Fourier transform of the field at the exit surface of the sample, which is physically similar to the far field diffraction pattern obtained with the original Z-scan method. Another goal of this lens is to produce an image of the object plane allowing to charac- terize accurately the profile of the beam at the entry.
Our numerical procedure in the Fourier domain has the advantage to reduce the computing time because calculations are done using fast Fourier transform. By using the optical transfer function related to the free- space propagation over a finite distance and considering the phase transformations, due to lenses L1 and L2, we can simulate the propagation of the beam from the object to the image plane taking into account the NL medium positioned at each motor step of the Z-scan procedure.
The influence of the transmittance of the NL medium is considered by means of the appropriate function (see Appendices 1 and 2), possibly using Newton’s method.
Finally, a fitting of the Z-scan profile allows to obtain the NL characteristic of the medium by comparing the numerical data with the experimental ones.
We used a 1-mm-thick cell filled with CS2excited by a Nd:YAG laser delivering linearly polarized picosecond pulses atk=1.064lm (17 ps) and 532 nm (12 ps) with a repetition rate\1 Hz. In the image plane, we use a cooled (-30°C) 1,00091,018 pixels CCD camera with a fixed linear gain. Each pixel 12912lm2 shows 4,095 gray
levels. The input intensity I0can be varied by means of a half-wave plate and a Glan prism (not shown in this figure) in order to maintain linear polarization. Two sets of acquisitions are performed. The first one is in the NL regime and the second one in the linear regime (by reducing the incident energy of the laser under the detec- tion limit of the joulemeter). The linear acquisitions are necessary to remove the diffraction, diffusion and/or imperfection effects due to sample inhomogeneities from the NL ones. Moreover, an image of the entry plane of the 4fsystem allows to define the object that will be considered in the simulation. The sensitivity of the setup depends on different parameters such as the object profile and the wavelength. In our procedure, this sensitivity is calculated for each set of acquisitions taking into account the real profile, which allows the measurement to be done with higher accuracy, even for relatively large absorption and phase shift when the signal is no more linear with the beam waist relative variation (BWRV) (for more detail on the numerical procedure see [8,13]). It is important to note that we perform absolute measurements [14] avoiding an intensity calibration dependant on a material. The main source of uncertainty of the measurements presented in the paper comes from the absolute measurement of the energy delivered by the pulsed laser; the accuracy of our calibrated joulemeter is about 10 % at the considered wavelengths.
3.1 Nonlinear absorption coefficients
The carbon disulfide is considered as a reference material for NL characteristic measurements. It is well known that CS2does not exhibit linear absorption coefficientaor 2PA coefficientbat 532 nm. However, with increasing incident intensity, the phenomenon of 3PA becomes predominant at 800 nm [15]. Thus, CS2is useful to test our method and to compare our results on the fifth-order coefficient with that obtained in [15] despite the difference between the incident wavelengths and temporal lengths. We performed an open- aperture Z-scan normalized transmittance to evaluate the NL absorption using a peak intensity at the focus inside the cellI0=25 GW/cm2. The radius of the circular object at the entry was 1.7 mm producing a beam waist in the focal plane of L1(f1=20 cm) about 19lm. The empty circles in Fig.2correspond to the experimental data. We realized a numerical fitting using Newton’s method (see above) with two different analytical expressions related to lines six and seven in Appendix1and corresponding to the cases of pure 2PA or 3PA. As we can see in Fig.2, considering only 2PA we obtain b=(8.5±0.9)910-12m/W (blue dot- ted line), but the best fit is found for 3PA with coefficient c=(9.3±1.9)910-26m3/W2 (solid red line). In Ref.
[15], the authors found a value ofc=1.37910-27m3/W2 using 120 ps pulses at 800 nm which is in good agreement
CCD
f1 f1 f2
L1 L2
L3
M1
BS1 NLM
BS2
f2
Z y x
M2
Image plane Object
plane
0 +Z -Z
Fig. 1 The 4fimaging system. The NL material (NLM) is moved around the focal plane (Z=0). The labels refer to lenses (L1,L2and L3), beam splitters (BS1andBS2) and mirrors (M1andM2). The object at the entry is a circular aperture. The image is acquired by a CCD camera
with the value that we measure considering the difference in the experimental parameters. Moreover, one of the CS2 absorption bands is approximately centered at 200 nm (see Fig.1a in Ref. [6]), that is approximately k/3 for k=532 nm. The proximity with this resonance band leads to increasevð5Þ/cwhich is supported by the fitting of the 3PA (solid red line in Fig.2), more close to the experimental data than the 2PA one.
We have performed the same experiment with CS2 at k=1.064lm (see Fig.3). The peak intensity at the focus inside the cell wasI0=65 GW/cm2. The radius of
the circular object at the entry was 1.9 mm producing a beam waist in the focal plane of L1 about 35lm. The solid red line shows the numerical fitting using the same formula as previously, leading to c=(4.6±0.9)9 10-27m3/W2, while the blue dotted line corresponds to the numerical fitting using formula line six, in Appendix 1, leading to b=(1.7±0.2)910-12m/W. It appears that the solid red line fits better the experimental points showing that CS2 does not exhibit significant 2PA characteristic but mainly 3PA. Note that the Z-scan profiles of the 2PA and 3PA are unfortunately too close, inducing a lot of errors in the measured values found in the literature. When accurately defined, the object profile is a mandatory parameter allowing to distinguish without any ambiguity between 2PA and 3PA open-aperture normalized Z-scan transmittance. Moreover, the method needs approximately 1 min of computing time for 61 Z-positions with less than 50 iterations for each point of the curves.
In the next paragraph, we will apply our procedure to determine both third-order and fifth-order NL refractive indices.
3.2 Nonlinear refractive index
Following the D4rmethod described in details in [8], we measured the BWRV using the first- and the second-order moments of the image profile versusZ, the position of the sample in the focus. In both cases, k=532 and k =1,064 nm, we measured n2 and b at low intensity, where the fifth-order contribution is insignificant. The experimental traces corresponding to these measurements are not shown here. The obtained results are n2= (1.5±0.3)910-18m2/W, b\0.2 cm/GW at I0=1.6 GW/cm2 for k =532 nm and n2=(4.5±1.3)9 10-19m2/W, b\0.05 cm/GW at 4.5 GW/cm2 for k =1,064 nm. Note that the sensitivity decreases with increasing wavelengths leading to higher intensity in the infrared measurement.
Then, these measured values were fixed for the second set of acquisitions performed at higher intensity to deter- mine fifth-order coefficients. According to [7], for the NL phase shift measurements, the medium can be regarded as thin (in which self-refraction process is considered as
‘‘external self-action’’), provided that: Du(0)ZR/L, whereZRis the Rayleigh range of the beam and Du(0) is the peak phase shift at the focus plane (Z=0). To satisfy this condition, taking into account our experimental parameters,Du(0) must be less thanpapproximately.
At k=532 nm, the images are acquired using I0=25 GW/cm2, which corresponds to the previous open- aperture Z-scan transmittance shown in Fig.2. We Fig. 2 Open-aperture Z-scan normalized transmittance (empty cir-
cles) of a 1-mm-thick cell filled with CS2measured atk=532 nm with I0=25 GW/cm2. The solid line (red) shows the numerical fitting considering only 3PA wherec=(9.3±1.9)910-26m3/W2, the dotted line (blue) shows the fitting considering only 2PA:
b=(8.5±0.9)910-12m/W
Fig. 3 Open-aperture Z-scan normalized transmittance (empty cir- cles) of a 1-mm-thick cell filled with CS2measured atk=1,064 nm withI0=65 GW/cm2. Thered solid lineshows the numerical fitting with c=(4.6±0.9)910-27m3/W2 (pure 3PA) while the blue dotted linewithb=(1.7±0.2)910-12m/W (pure 2PA)
measured the BWRV and the results are shown in Fig.4 (empty circles). In order to take into account the third- and fifth-order contributions at this relatively high intensity, the analytical formula line six in Appendix2 is used to fit the data where all parameters, exceptn4, are already known.
Using the values measured at low intensity (n2=(1.5± 0.3)910-18m2/W, b=0) and taking into account the value obtained above for the 3PA coefficient,c=(9.3± 1.9)910-26m3/W2, the solid (red) line shown in Fig.4is obtained for n4=(1.2±0.3)910-32 m4/W2. Then, in
order to show the error that could result we compute the BWRV assuming that the NL refraction is limited to third-order susceptibility only. The blue dotted line in Fig.4 is obtained with n2=(2.7±0.3)910-18m2/W using the 2PA coefficient found above,b=(8.5±0.9)9 10-12m2/W.
At k=1.064lm, the BWRV of the images acquired and corresponding to the previous open-aperture Z-scan transmittance shown previously in Fig.3 were processed and the results appear as empty circles in Fig.5. We used the same formulas as in Fig.4. When considering together third- and fifth-order nonlinearities, we obtainn4=(2.2± 0.4)910-33m4/W2 using the above result for c= (4.6±0.9)910-27m3/W2and as before the value ofn2 obtained through the low intensity measurement (n2=(4.5±1.3)910-19m2/W,b=0). The solid (red) line shown in Fig.5 represents the fitting with these parameters. At this relatively high intensity level, taking into account the significant 3PA of CS2 (related to the imaginary part of v(5)), it is likely to consider a response following a combination ofn2andn4. The blue dotted line represents the fitting of the experimental data consider- ing only third-order contribution with the parameters b=(1.7±0.2)910-12m/W and n2=(1.4±0.2) 910-18m2/W. In both cases, the agreement between the fits and the experimental data is very good, showing again that confusion could appear when considering data at high intensity only. One way to avoid any ambiguity is to perform a D4r-Z-scan measurement with a lower intensity. At 4.6 GW/
cm2, we have foundn2=(4.5±1.5)910-19m2/W, which is a value in very good agreement with other measured values published in the literature by different groups [14,16] and [17]
the one found here with our fitting. If we consider that the CS2 response at high intensity is only due ton2contribution, one can overestimate then2values for both considered wavelengths.
Our results are summarized in Appendix3considering third- and fifth-order responses.
Furthermore, it is common and usual to calibrate the measurement system using the n2 value of CS2 because of its high nonlinearities showing a good contrasted signal-to-noise ratio. Besides the fact that this material has different responses at different excitations depending on the pulse width, wavelength or polarization, here we point out a relatively high dependence on the intensity.
We should avoid the calibration with this material even for relative measurement or clearly indicate the level of intensity that has been used for the calibration. If the experimental setup is sensitive and the signal high enough, it is better to consider fused silica for calibra- tion because (1) it shows an ultrafast (femtosecond) response, unlike CS2 which has dual decay time in the femtosecond and picosecond regimes and (2) needs Fig. 4 Beam waist relative variation (empty circles) versusZof a 1–
mm-thick cell filled with CS2 measured at k=532 nm with I0=25 GW/cm2. Theblue dotted lineis obtained considering only v(3) contribution: b=(8.5±0.9)910-12m/W and n2=(2.7± 0.3)910-18m2/W. The solid line(red) is found with:c=(9.3± 1.9)910-26m3/W2, n2=(1.5±0.3)910-18m2/W and n4= (1.2±0.3)910-32m4/W2
Fig. 5 Beam waist relative variation (empty circles) of a 1-mm-thick cell filled with CS2 measured atk=1.064lm with I0=65 GW/
cm2. The blue dotted lineis obtained considering onlyv(3) contri- bution: b=(1.7±0.2)910-12m/W and n2=(1.4±0.2)9 10-18m2/W. The red solid line is found for: c=(4.6±1.9)9 10-27m3/W2, n2=(4.5±1.3)910-19m2/W and n4=(2.2± 0.4)910-33m4/W2
relatively higher intensity to excite the fifth-order response [18].
4 Conclusion
In summary, we have found the analytical solution of the propagation distance and the phase shift inside the non- linear medium versus the output intensity in the case of third- and fifth-order NL susceptibilities. We introduced a numerical inversion by means of Newton’s method. The solutions are given in all the cases whatever is the inde- terminacy due to vanishing coefficients. Combined with D4r-Z-scan method in the case of CS2, this analysis allowed us to measure the third- and fifth-order NL optical susceptibilities.
Appendix 1: Solution forz(I) See Table1.
Appendix 2: Solution for the nonlinear phase shiftDu(I) See Table2.
Appendix 3: Summary of the measured nonlinear coefficients
See Table3.
Table 1 Solution of Eq. (1) depending on the vanishing parameters of the linear and nonlinear absorption coefficients Value of each coefficient Solution to Eq. (1)
1 a,b,c=0 andb2[4ac
z¼ ln
IL I0 cXþXþ ln
ILX I0X
cXðXXþÞ ln
ILXþ I0X
cXþðXXþÞ
2 4
3 5
2 a,b,c=0 andb2\4ac
z¼1 a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 4cab2
p tan1 2cILþb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4cab2 p
!
tan1 2cI0þb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4cab2 p
!
" #
(
þ1
2ln I02aþbILþcI2L IL2aþbI0þcI02
" #)
3 c=0 anda,b=0 z¼1alnII0ðbIþaÞ
LðbI0þaÞ
h i
4 b=0 anda,c=0
z¼2a1ln cI
2 Lþa
ð ÞI20 cI02þa
ð ÞI2L
5 a=0 andb,c=0 z¼ bc2ln II0ðcILþbÞ
LðcI0þbÞ
h i
IbI0IL
LI0
n o
6 a,c=0 andb=0 z¼IbI0IL
LI0
7 a,b=0 andc=0 z¼2cII20I22L
LI20
Table 2 Solution of Eq. (2) depending on the vanishing parameters of the linear and nonlinear absorption coefficients Value of each coefficient Solution to Eq. (2)
1 a,b,c=0 andb2[4ac Du¼cXk
þX
ð Þ ðn2þn4XþÞln IILXþ
0Xþ
þðn2þn4XÞln IILX
0X
h i
2 a,b,c=0 andb2\4ac
Du¼k c
bn42cn2
ð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4cab2
p tan1 2cILþb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4cab2 p
!
tan1 2cI0þb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4cab2 p
!
" #
(
þn4
2ln aþbILþcI2 aþbI0þcI20
3 c=0 anda,b=0 Du¼ k nb2anb24
ln bIbILþa
0þa
þnb4ðILI0Þ
h i
4 b=0 anda,c=0
Du¼ k n2c4ln cIcI2L2þa 0þa
þpn2ffiffiffiffiac tan1 ffiffi
c a
q IL
tan1 ffiffi
c a
q I0
h i
n o
5 a=0 andb,c=0 Du¼ knb2ln IIL
0 cn2bncb 4ln cIcILþb
0þb
6 a,c=0 andb=0 Du¼kb n2ln IIL
0 þn4ðILI0Þ
h i
7 a,b=0 andc=0 Du¼kc n2ILII0
LI0 þn4ln IIL
0
h i
References
1. G.L. Wood, W.W. Clark III, M.J. Miller, G.J. Salamo, E.J. Sharp, J. SPIE Proc.1105, 154–180 (1980)
2. H. Shim, M. Liu, C. Hwangbo, G.I. Stegeman, Opt. Lett.23, 430 (1998)
3. G. Lenz, J. Zimmermann, T. Katsufuji, M.E. Lines, H.Y. Hwang, S. Spalter, R.E. Slusher, S.W. Cheong, J.S. Sanghera, I.D. Ag- garwal, Opt. Lett.25, 254 (2000)
4. F. Smektala, C. Quemard, L. Leneindre, J. Lucas, A. Barthelemy, C. De Angelis, J. Non Cryst. Solids239, 139 (1998)
5. V. Besse, H. Leblond, D. Mihalache, B.A. Malomed, Phys. Rev.
E87, 012916 (2013)
6. E.L. Falca˜o-Filho, C.B. de Arau´jo, G. Boudebs, H. Leblond, V.
Skarka, PRL110, 013901 (2013)
7. M. Sheik-Bahae, A.A. Said, T.H. Wei, D. Hagan, E.W. Van Stryland, IEEE J. Quant. Electron.26, 4 (1990)
8. G. Boudebs, V. Besse, C. Cassagne, H. Leblond, C.B. de Arau´jo, Opt. Lett.38, 13 (2013)
9. G. Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F.
Smektala, F. Sanchez, Opt. Commun.232, 417 (2004) 10. S. Cherukulappurath, J.L. Godet, G. Boudebs, J. Nonlinear Opt.
Phys. Mater.14, 1 (2005)
11. G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F.
Smektala, F. Sanchez, Opt. Commun.219, 427–433 (2003) 12. K. Fedus, G. Boudebs, Opt. Commun.292, 140 (2013) 13. G. Boudebs, K. Fedus, C. Cassagne, H. Leblond, Appl. Phys. Let.
93, 021118 (2008)
14. G. Boudebs, K. Fedus, J. Appl. Phys.105, 103106 (2009) 15. D.G. Kong, Q. Chang, H.A. Ye, Y.C. Gao, Y.X. Wang, X.R.
Zhang, K. Yang, W.Z. Wu, Y.L. Song, J. Phys. B At. Mol. Opt.
Phys.42, 065401 (2009)
16. M. Falconieri, G. Salvetti, Appl. Phys. B Lasers Opt.69, 133–136 (1999)
17. S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E.
Koudoumas, X. Michaut, Chem. Phys. Lett.369, 318–324 (2003) 18. M. Grehn, T. Seuthe, W.J. Tsai, M. Ho¨fner, A.W. Achtstein, A.
Mermillod-Blondin, M. Eberstein, H.J. Eichler, J. Bonse, Opt.
Mater. Express3(12), 2132–2140 (2013) Table 3 Summary of the third- and the fifth-order measured non-
linear coefficient values for CS2in the picosecond regime with lin- early polarized light
Wavelength k=532 nm k=1,064 nm
Linear absorptiona (m-1)
0 0
Two-photon absorptionb(m/W)
0 0
Third-order nonlinear refractive indexn2 (m/W2)
(1.5±0.3)910-18 (4.5±1.3)910-19
Three-photon absorptionc(m3/W2)
(9.3±1.9)910-26 (4.6±0.9)910-27 Fifth-order nonlinear
refractive indexn4 (m4/W2)
(1.2±0.3)910-32 (2.2±0.4)910-33
